Chapter 1.2, Problem 55E

To determine

To sketch : The graph of the function $y=f(x)$ that satisfies the stated condition.

Answer to Problem 55E

The graph is shown:

  

Explanation of Solution

Given information: The limits are

  $\begin{array}{c}\underset{x\to 1}{\lim }f(x)=2\\ \underset{x\to 5^{-}}{\lim }f(x)=\infty \\ \underset{x\to 5^{+}}{\lim }f(x)=\infty \\ \underset{x\to \infty }{\lim }f(x)=-1\\ \underset{x\to 2^{+}}{\lim }f(x)=-\infty \\ \underset{x\to 2^{-}}{\lim }f(x)=\infty \\ \underset{x\to \infty }{\lim f(x)}=0\end{array}$

Graph:

The required graph is shown below:

  

Hence, assume that $f(1)=2$ to make the graph simpler so plot the point $(1,2)$

  $\underset{x\to c^{+}}{\lim }f(x)=\pm \infty$ or $\underset{x\to c^{-}}{\lim }f(x)=\pm \infty$ there is a vertical asymptote at $x=c$ . Therefore there must be a vertical asymptote at $x=5$ and $x=-2$ .For $x=-2,$ Given that the limit is positive infinity, the graph must ascend on the left side of the vertical asymptote and descend on the right side. The graph must increase on both sides for $x=5$ .

  $\underset{x\to \infty }{\lim }f(x)=c$ or $\underset{x\to -\infty }{\lim }f(x)=c$ ,there is a horizontal asymptote $y=c$ . As a result, the horizontal asymptotes for the right side of the graph are $y=-1$ and the the asymptotes for the left are $y=0$ .